How to fix ugly bevel in inner corner?
Your edge theory is right. The face the corner is in is something called an N-gon. It is usually discouraged due to several problems with its internal handling.

When you create that edge using F, the created edge does not separate the face, so the N-gon stays. In order to actually cut the face, use the Knife Tool K in Edit Mode. Select one of the vertices. You will see there is now a square and a purple line. Now click on the other vertex and hit Enter. Now, the N-gon is separated.

The N-gon gets split into more N-gons, so you might want to use the Knife Tool more to separate it into quads and triangles (which is good topology).

Your corner is now fixed.

Documentation on Knife Tool: https://docs.blender.org/manual/en/latest/modeling/meshes/editing/subdividing/knife.html